Finally, we note that one can find the joint probability distribution of \(Z_t\)  and \(Z_{t+1}\)  by formulating an optimal transport problem [1]} to minimize \(\mathbb {E}\left[\left\Vert Z_{t+1} - Z_t\right\Vert _{1,w^{\prime }}\right]\) . This approach may lead a smaller expected update cost, but it requires to explicitly consider all possible values of \(\mathbf {Z}_t\)  and \(\mathbf {Z}_{t+1}\) , that may be as large as \(|\mathcal {Z}| = {N \atopwithdelims ()k}\) . Our algorithm instead efficiently builds the support of these random variables at each step to have size at most linear in \(N\) .
